Translate each of these nested quantifications into an Engli

Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers. (4 points)

a) xy(xy = y)

b) xy(((x < 0) (y < 0)) (xy > 0))

c) xy((x² > y) (x < y))

d) xyz (x + y = z)

Solution

a) xy(xy = y)

This says that there exists a real number x such that for every real number y, the product of x and y equals y.

b)  xy(((x < 0) (y < 0)) (xy > 0))

For all negative values of x and y, the product is a positive number.

c) xy((x² > y) (x < y))

There exist some x and some y such thatx is less than y and square of z is greater than y.

d) xyz (x + y = z)

For all x and y, there exist a z such that z is sum of x and y.